Fundamental groups of log Calabi-Yau surfaces

In this article, we study the orbifold fundamental group ${\pi }_1^{\mathrm{o}\mathrm{r}\mathrm{b}}(X,\mathrm{\Delta })$ of a Calabi–Yau pair $(X,\mathrm{\Delta })$ with log canonical singularities. We conjecture that the orbifold fundamental group ${\pi }_1^{\mathrm{o}\mathrm{r}\mathrm{b}}(X,\mathrm{\Delta })$ of a $n$-dimensional log Calabi–Yau pair admits a normal solvable subgroup of rank at most $2n$ and index at most $c(n)$. We prove this conjecture in the case that $n=2$. More precisely, for a log Calabi–Yau surface pair $(X,\mathrm{\Delta })$ we show that ${\pi }_1^{\mathrm{o}\mathrm{r}\mathrm{b}}(X,\mathrm{\Delta })$ is the extension of a nilpotent group of length at most $2$ and rank at most $4$ by a finite group of order at most $7200$. We also show that the bounds on the nilpotency length, rank, and order of the finite group quotient in this result are sharp. Finally, we provide some necessary criteria for a log Calabi–Yau surface $(X,\mathrm{\Delta })$ to have an infinite, or a non virtually abelian orbifold fundamental group.